Connecting Polygonizations via Stretches and Twangs

TL;DR

This paper demonstrates that the space of polygonizations of a fixed planar point set is connected through a sequence of simple atomic moves called stretches and twangs, which could facilitate random polygon generation.

Contribution

It introduces a new framework of moves connecting polygonizations, enabling systematic exploration of polygon spaces for the first time.

Findings

Connected polygonizations via O(n^2) moves

Moves consist of stretches and twangs

Potential for random polygon generation

Abstract

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n^2) ``moves'' between simple polygons. Each move is composed of a sequence of atomic moves called ``stretches'' and ``twangs''. These atomic moves walk between weakly simple ``polygonal wraps'' of S. These moves show promise to serve as a basis for generating random polygons.